$$AppliedStatistics$$$$AppliedStatistics$$AppliedStatisticsApplied\,Statistics$$AppliedStatistics$$

Curve Fitting by the Method of Least Squares

Linear Curve Fitting

• •$m$$m$mm$m$ is the slope of the line, representing the rate of change of $y$$y$yy$y$ with respect to $x$$x$xx$x$.
• •$b$$b$bb$b$ is the y-intercept, the value of $y$$y$yy$y$ when $x$$x$xx$x$ is 0.

Solving Linear Curve Fitting by Simplified Approach

• •$\sum x$$\sum x$sum x\sum x$\sum x$: The sum of all $x$$x$xx$x$-values in the dataset.
• •$\sum y$$\sum y$sum y\sum y$\sum y$: The sum of all $y$$y$yy$y$-values in the dataset.
• •$\sum xy$$\sum xy$sum xy\sum xy$\sum xy$: The sum of the products of each $x$$x$xx$x$-value and its corresponding $y$$y$yy$y$-value.
• •$\sum x^2$$\sum x^2$sumx^(2)\sum x^2$\sum x^2$: The sum of the squares of all $x$$x$xx$x$-values in the dataset.

Polynomial Curve Fitting

Nonlinear Curve Fitting

Introduction to Hypothesis Testing

Key Components

1. $\mathbf{\text{Null Hypothesis (}}{H}_0\mathbf{\text{):}}$$\mathbf{\text{Null Hypothesis (}}{H}_0\mathbf{\text{):}}$"Null Hypothesis ("(H_(0))"):"\textbf{Null Hypothesis (\(H_0\)):}$\mathbf{\text{Null Hypothesis (}}{H}_0\mathbf{\text{):}}$ This is the initial assumption that there is no significant difference or effect. It represents the status quo and is usually denoted by $H_0$$H_0$H_(0)H_0$H_0$.
2. $\mathbf{\text{Alternative Hypothesis (}}{H}_a\mathbf{\text{):}}$$\mathbf{\text{Alternative Hypothesis (}}{H}_a\mathbf{\text{):}}$"Alternative Hypothesis ("(H_(a))"):"\textbf{Alternative Hypothesis (\(H_a\)):}$\mathbf{\text{Alternative Hypothesis (}}{H}_a\mathbf{\text{):}}$ This is the claim or statement we want to investigate and determine if there is enough evidence to support. It opposes the null hypothesis and is denoted by $H_a$$H_a$H_(a)H_a$H_a$.

Types of Hypotheses

• •$\mathbf{\text{One-Tailed (or One-Sided) Hypothesis:}}$$\mathbf{\text{One-Tailed (or One-Sided) Hypothesis:}}$"One-Tailed (or One-Sided) Hypothesis:"\textbf{One-Tailed (or One-Sided) Hypothesis:}$\mathbf{\text{One-Tailed (or One-Sided) Hypothesis:}}$ This type of hypothesis test focuses on either a positive or negative effect. It is denoted as $H_a:\mu >{\mu }_0$$H_a:\mu >{\mu }_0$H_(a):mu > mu_(0)H_a: \mu > \mu_0$H_a:\mu >{\mu }_0$ (right-tailed) or $H_a:\mu <{\mu }_0$$H_a:\mu <{\mu }_0$H_(a):mu < mu_(0)H_a: \mu < \mu_0$H_a:\mu <{\mu }_0$ (left-tailed), where $\mu$$\mu$mu\mu$\mu$ is the population mean and ${\mu }_0$${\mu }_0$mu_(0)\mu_0${\mu }_0$ is the hypothesized value.
• •$\mathbf{\text{Two-Tailed Hypothesis:}}$$\mathbf{\text{Two-Tailed Hypothesis:}}$"Two-Tailed Hypothesis:"\textbf{Two-Tailed Hypothesis:}$\mathbf{\text{Two-Tailed Hypothesis:}}$ This type of hypothesis test is more general and looks for any significant difference, regardless of the direction. It is denoted as $H_a:\mu \ne {\mu }_0$$H_a:\mu \ne {\mu }_0$H_(a):mu!=mu_(0)H_a: \mu \neq \mu_0$H_a:\mu \ne {\mu }_0$.

Steps for Hypothesis Testing

1. $\mathbf{\text{Formulate Hypotheses:}}$$\mathbf{\text{Formulate Hypotheses:}}$"Formulate Hypotheses:"\textbf{Formulate Hypotheses:}$\mathbf{\text{Formulate Hypotheses:}}$ Define the null and alternative hypotheses based on the research question.
2. $\mathbf{\text{Select Significance Level (}}\alpha \mathbf{\text{):}}$$\mathbf{\text{Select Significance Level (}}\alpha \mathbf{\text{):}}$"Select Significance Level ("alpha"):"\textbf{Select Significance Level (\(\alpha\)):}$\mathbf{\text{Select Significance Level (}}\alpha \mathbf{\text{):}}$ The significance level is the probability of rejecting the null hypothesis when it is actually true. Common choices include 0.05 (5%) or 0.01 (1%).
3. $\mathbf{\text{Choose a Test Statistic:}}$$\mathbf{\text{Choose a Test Statistic:}}$"Choose a Test Statistic:"\textbf{Choose a Test Statistic:}$\mathbf{\text{Choose a Test Statistic:}}$ The choice of test statistic depends on the type of data and the specific hypothesis test.
4. $\mathbf{\text{Calculate the Test Statistic:}}$$\mathbf{\text{Calculate the Test Statistic:}}$"Calculate the Test Statistic:"\textbf{Calculate the Test Statistic:}$\mathbf{\text{Calculate the Test Statistic:}}$ Using the sample data, compute the value of the test statistic.
5. $\mathbf{\text{Determine the Critical Region:}}$$\mathbf{\text{Determine the Critical Region:}}$"Determine the Critical Region:"\textbf{Determine the Critical Region:}$\mathbf{\text{Determine the Critical Region:}}$ This is the region of extreme values of the test statistic that leads to the rejection of the null hypothesis.
6. $\mathbf{\text{Compare Test Statistic with Critical Value:}}$$\mathbf{\text{Compare Test Statistic with Critical Value:}}$"Compare Test Statistic with Critical Value:"\textbf{Compare Test Statistic with Critical Value:}$\mathbf{\text{Compare Test Statistic with Critical Value:}}$ If the test statistic falls within the critical region, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

Common Test Statistics

• •$\mathbf{\text{Z-Test:}}$$\mathbf{\text{Z-Test:}}$"Z-Test:"\textbf{Z-Test:}$\mathbf{\text{Z-Test:}}$ Used when the population standard deviation ($\sigma$$\sigma$sigma\sigma$\sigma$) is known, and the sample size is sufficiently large.
  
  Formula: $Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$$Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$Z=(( bar(X))-mu)/(sigma//sqrtn)Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}$Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$
• •$\mathbf{\text{T-Test:}}$$\mathbf{\text{T-Test:}}$"T-Test:"\textbf{T-Test:}$\mathbf{\text{T-Test:}}$ Used when the population standard deviation ($\sigma$$\sigma$sigma\sigma$\sigma$) is unknown or the sample size is small ($n<30$$n<30$n < 30n < 30$n<30$).
  
  Formula (for one-sample t-test): $t=\frac{\overline{X}-\mu }{s/\sqrt{n}}$$t=\frac{\overline{X}-\mu }{s/\sqrt{n}}$t=(( bar(X))-mu)/(s//sqrtn)t = \frac{\bar{X} - \mu}{s/\sqrt{n}}$t=\frac{\overline{X}-\mu }{s/\sqrt{n}}$, where $s$$s$ss$s$ is the sample standard deviation.
• •$\mathbf{\text{Chi-Square Test (}}{\chi }^2\mathbf{\text{):}}$$\mathbf{\text{Chi-Square Test (}}{\chi }^2\mathbf{\text{):}}$"Chi-Square Test ("(chi^(2))"):"\textbf{Chi-Square Test (\(\chi^2\)):}$\mathbf{\text{Chi-Square Test (}}{\chi }^2\mathbf{\text{):}}$ Used for categorical data to test independence or goodness of fit.
  
  Formula: ${\chi }^2=\sum \frac{(O-E{)}^2}{E}$${\chi }^2=\sum \frac{(O-E{)}^2}{E}$chi^(2)=sum((O-E)^(2))/(E)\chi^2 = \sum \frac{(O - E)^2}{E}${\chi }^2=\sum \frac{(O-E{)}^2}{E}$, where $O$$O$OO$O$ is the observed frequency and $E$$E$EE$E$ is the expected frequency.
• •$\mathbf{\text{F-Test:}}$$\mathbf{\text{F-Test:}}$"F-Test:"\textbf{F-Test:}$\mathbf{\text{F-Test:}}$ Used to compare the variance of two or more samples.
  
  Formula (for two-sample F-test): $F=\frac{s_1^2}{s_2^2}$$F=\frac{s_1^2}{s_2^2}$F=(s_(1)^(2))/(s_(2)^(2))F = \frac{s_1^2}{s_2^2}$F=\frac{s_1^2}{s_2^2}$, where $s_1^2$$s_1^2$s_(1)^(2)s_1^2$s_1^2$ and $s_2^2$$s_2^2$s_(2)^(2)s_2^2$s_2^2$ are the sample variances.